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Interactive Visualization across the Curriculum

Interactive Visualization across the Curriculum. Matthias Kawski Department of Mathematics Arizona State University Tempe, Arizona U.S.A. Thanks for generous support by. Department of Mathematics Center for Research in Education of Science, Mathematics, Engineering, and Technology

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Interactive Visualization across the Curriculum

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  1. Interactive Visualizationacross the Curriculum Matthias Kawski Department of Mathematics Arizona State University Tempe, Arizona U.S.A. http://math.la.asu.edu/~kawskikawski@asu.edu

  2. Thanks for generous support by Department of Mathematics Center for Research in Education of Science, Mathematics, Engineering, and Technology Arizona State University INTEL Corporation through grant 98-34 National Science Foundation through the grants DUE 97-52453 Vector Calculus via Linearization: Visualization and Modern Applications DMS 00-72369 Algebra and Geometry of Nonlinear Control Systems EEC 98-02942 Engineering Foundation Coalition http://math.la.asu.edu/~kawskikawski@asu.edu

  3. Vector Calculus Differential Equations Linear Algebra Calculus I, II, III Adv. Engineering Math [[Advanced Calculus]] [How to write a proof.] Differential Geometry Complex Analysis Partial Diff. Equations Techn. in Classroom Advanced Math via CAS “Across” the Curriculum Moved practically all courses aggressively into predominantly electronic format, emphasizing interactive visualization, exploration and experimentation – but also proving via programming or via CAS. Last 5 years: http://math.la.asu.edu/~kawskikawski@asu.edu

  4. Everything is on the WWW. . . . . . the key is to use it intelligently! • Reconsider purpose/objectives of each course... • Reconsider selection of content / specific topics • Reconsider choice of delivery methods • “Inquiry-based learning”Doing math = experiment, make observations, conjecture, further test, formulate theorem, prove  definition, axiomatize…. • Concentrate on core topics, and study these in depthThere are very few fundamental concepts, emphasize “coherence”, Build rich “rooted” concept (“procept”) images (often w/ visual core!!!) . . . . and remember them for life (as opposed to: memorize lots of formulas for next exam only) • Efficiency and effectivenessIntense, highly interactive sessions via modern software tools. Continuous assessment & evaluation, permitting and relying on WWW... http://math.la.asu.edu/~kawskikawski@asu.edu

  5. E.g. choice between CAS and JAVA Selected parameters Coming next: Selected examples from • Differential geometry • Complex Analysis • Vector Calculus http://math.la.asu.edu/~kawskikawski@asu.edu

  6. JAVA 2 or CAS? Complex analysis • Convergence of Laurent series • Zooming on essential singularities • Winding numbers & branch cuts • Conformal mappings http://math.la.asu.edu/~kawskikawski@asu.edu

  7. Differential Geometry, Gauss curvature After proving (!) e.g. the Theorema Egregium withMAPLE ((-- is that really doing mathematics?? askThurston!)), the real exciting math is just beginning! http://math.la.asu.edu/~kawskikawski@asu.edu

  8. JAVA 2 or CAS? Differential Geometry • Geodesic spheres, distance function • Conjugate points, focusing/curvature • Visualizing tensors http://math.la.asu.edu/~kawskikawski@asu.edu

  9. Vector calculus: Curl & divergence The central object of study in vector calculus. A horrible formula that few students remember beyond the next exam. Traditionally: almost exclusive use of algebraic symbols • little insight (one-sided, or fragmented, concept image) • major hurdle for re-entry students • invitation to further study higher math? http://math.la.asu.edu/~kawskikawski@asu.edu

  10. Curl & divergence  derivatives? ? http://math.la.asu.edu/~kawskikawski@asu.edu

  11. Curl: Coherence*)or fragmentation?*) and connections, i.e. hyperlinked architecture in brain just as on WWW http://math.la.asu.edu/~kawskikawski@asu.edu

  12. Compartmentalization /Fragmentation ! Complex Analysis Linear Algebra Differential Equations http://math.la.asu.edu/~kawskikawski@asu.edu

  13. JAVA - Vector field analyzer • Interactively zoom (move the lens!) to see derivatives of vector fields, curl and divergence, ..... • Switch to flows – connect VC w/ DEs • Coming soon:Line / flux integrals http://math.la.asu.edu/~kawskikawski@asu.edu

  14. This was just a “teaser”! Also p.65:Friday 1:00 – 3:00 p.m. NSF – DUE CCLI poster session Sheraton, Pontchartrin A 3rd floor • Course materials • JAVA applets • CAS worksheets • past conference presentations (mostly .ppt ) • (p)reprints of published articles are, of course, all on the WWW ...... just use your favorite search engine, and look for “kawski” http://math.la.asu.edu/~kawski http://math.la.asu.edu/~kawskikawski@asu.edu

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